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AD-A13 316 APPROXIMATIONMETHDS INMULTDIMENSIONAL FILER DESIGN 1/AND RELATED PROBL.U) PITSBURGH UNIV PA DEPTOELECTRICAL ENDINEERING N K BOSE 21 MAR 83 F2DNCLASSIFED AFOSR-TR-83-0661 AFOSR-78-3542 FG1/ l
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SECURITY CLASSIFICATION OF THIS PAGE M.ien DeeaEnt.e.d)REPORT DPAGE READ INS rRUCTIONS
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AFOSR.TR. 83 -0 6 61 Wv,3' __.2/__(6,4. TITLE (amd Subtitle) S. TyPE OF REPORT & PER:OD COVE'tt 0
Approximation Methods in Multidimensional Filter Final (1-1-75 to 1-31-83)Design and Related Problems Encountered in Multi-dimensional System Design 6. PERFORMING O1. REPORT N..'ABER
7. AUTHOP(a) S. CONTRACT OR GRANT NUMBER(A;
Professor N. K. Bose AFOSR 78-3542
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT
PRO.ECf. TASKy of Psbuh AREA & WORK UNIT NUMBERS
Department of Electrical Engineering 6C F
Pittsburgh, PA 15261)I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE
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1g. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on revere* side It neces.a. and Identify by block nu mber)
0L Multidimensional Systems Spatio-Temporal FilteringRational Approximants Primitive FactorizationStability Irreducible polynomialsStabilization Matrix Stieltjes seriesRobustness Digital Filters over finite field-_J
20. ABSTRACT (Continue an ree. Aide If neeseeary and Identf by block number)
The research conducted contributes towards the development of a theory tcanalyze and design linear shift-invariant multivariable multidimensinnal dis-
s- crete and continucus systems. Recursive schemes to compute rational appro .i-Pmants to a power series in two variables having constant matrices for coef-ficients are developed. Approximants to special matrix power series areinvestigated and the properties of these approximants are delineated in astrictly mathematical setting and their implications are interpreted via
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physical reasonings. Algebraic procedures to test the approximants forstability are provided, and criteria for guaranteeing the invariance of prop-erties like positivity and stability under parameter changes or perturbationare advanced. Attention is directed throughout towards the reduction of
algebraic computational complexity. In the important problem of filterstabilization without appreciable change in the magnitude of the frequencyresponse, recent results on multiplicative computational complexity theory isexploited to demonstrate the feasibility of implementing efficiently a 2-D
discrete Hilbert transform. Criteria for 2-D rational approximants to bemaximally flat are obtained.
The structure and properties of impulse response arrays for discrete-space systems over a finite field of coefficients are investigated. Acomplete characterization of arrays whose properties are very close to the
properties of pseudorandom arrays, is given; instead of two distinct levelsof the discrete autocorrelation function, these characterized arrays areshown to have a maximum of three distinct levels, whose values can beexplicitly calculated from the denominator polynomial of the transfer function
generating the array.The feasibility of obtaining the primitive factorization of bivariate
polynomial matrices whose coefficients belong to an arbitrary but fixed fieldis demonstrated by the'development of an algorithm whose implementationrequire operations over the specified field and which also produces factorswith coefficients over the same field and not over some extension field. Th;iresult opens the door to several challenging practical problems in the realiza-tion theory of multidimensional systems. The primitive factorization result
is also used in the technique developed to stabilize 2-D discrete-space plants
using causal and weakly causal compensators.
E CL CLAT,: OP 't Ei ~~SECUPIT'f CLASS4ICATIONI OF r r ,Er'When Da),re Er
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TABLE OF CONTENTS
Items Pages
a. Cover and title page (form DD1473) 1-2
b. Research Objectives 4-11
c. Status of the Research 12-20
d. Publications in Technical Journals 21-23
e. List of the Professional Personnel Associated 24-25
With the Research Report
f. Interactions of Principal Investigator, Dr. N. K. Bose 26-30
g. Specific Applications Stemming from Research Report 31-34
h. Reprints and Preprints
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b. Research Objectives
(i) Rational Approximants
The topics here will be presented in a concise but clear manner so that
the reader encounters little difficulty in understanding the context and
technical merits. Most specifications are unrealizable from the physical
realizability standpoint and have to be approximated with respect to a chosen
error criterion. The approximation may be done either in the frequency or
spatial domains. It was recognized that the quality of the approximant must
be coupled to the facility or convenience with which the approximant can be
calculated. This necessitated the demand for attention to the scopes for
development or exploitation of fast algorithms; that is, an understanding of
the problem structure was considered crucial. At the same time, the approximation
technique was required to be fundamental and broad enough to support a wide
variety of applications. It was decided that the Pad6 approximation technique
would serve as an ideal nucleus from which the desired research directions and
objectives could ideally be reached.
The Pade approximant of order [L/M] to the formal power series t(z) =
Etk z is the rational function aL (z)/b(z), where aL(z) and bM(z) arek=O
polynomials of degree at most L and M respesctively such that
aL (z) L+M+lt(z) - b (z) O(z (1)
L+b M+zand O(z + ) in (1) is taken to mean that terms of any order less thanL+M+ I
z are missing. An approximant of any order, if it exists, is unique
and the problem of computing the approximant essentially consists of solving
a system of simultaneous linear equations
M [b M bM_ . . bl~t = -[tt+l tL+2 tt+M t (2)
where, HM is a (M x M) Hankel matrix obtained by setting ItLM+l tL-m42
Ll as the first row and (tL tL+l tL+Mlj t as the last column,
. . . . .. .. . . . . . . .- - U ... ..
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b k's in (2) are coefficients of polynomial bM(z) and 't' denotes the transpose
of a matrix. It is well known, however, that equations of the type in (2) occur
in the problem of partial realization of systems from prescribed input-output
maps [ l]. Recursive O(n 2 ) algorithms which exploit the Hankel structure of
the matrix HM, to solve for the system of equations (2) have been proposed [2].
It has also been noted recently [3] that these recursions fall under the class
of recursionsknown as Lanczos recursions [4 1. Furthermore, the same recurrence
formula occurs in the entirely different context of shift-register synthesis
and is called the Berlekamp-Massey relation [5]. One objective of the research
was to develop a scheme for recursive computation of the: Pads approximants,
when the coefficients of the power series t(z) are rectangular matrices.
A closer examination of the recurrence relations (in the scalar case)
reveals that the denominators of the successive menbers of the sequence of
approximants are connected by a three term recurrence formula similar to the
one followed by polynomials, which are orthogonal on a real interval. In fact,
it has been shown that, under some restrictive hypothesis, the sequence of
polynomials {zM bM(z-l)}M=dSsociated with the denominators of the sequence
{[(M-k)/M]}M 0 indeed form a set of orthogonal polynomials of the type just men-
tioned. Furthermore, the sequence of successive convergents of a continued
fraction expansion of a special type associated with t(z) is known to be the
same as the sequence of Padd approximants: [0/1], [1/0], (1/1], [2/1], [2/2],
• to t(z).
Also, the infinite series t(z) is known to be a Stieltjes series if certain
positive definiteness conditions on the Hankel matrices HM in (2) are satisfied.
A result of significant interest in the context of this discussion is that
the sequence of Pad6 approximants of certain orders to a Stieltjes series can
be shown to be realizable driving point impedance functions of electrical
networks consisting of two types (RC, RL, or LC) of elements. Proof for this
fact can be given by classical ladder type synthesis techniques or via a
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6
consideration of the Cauchy index of rational functions of one variable.
Another objective of the research was to investigate into the properties
of the "denominator" polynomial matrices with respect to any recurrence
relation they might obey. The Stieltjes series was planned for study in a
matrTk setting, especially with the objective of exploring into possible
links of the approximants to multiport network theory. The theory of matrix
polynomials, orthogonal over a real interval, was then natural to investigate,
especially with the possible availability of any physical insight from network
theory.
In view of the established importance of 2-D systems theory from both
theoretical and applications standpoints, the next objective was to attempt
to extend the Padg approximation scheme to the 2-D case by requiring the rational
approximant of prescribed degree t(z 1 ,z 2) to be such that Rij in (3) is
zero over a
t(zl, z - t(z, z ) = TO R z1 z2 (3)
finite set of points (i,j) having a chosen geometry in the two-dimensional
lattice. Two different geometries have been considered in [6] and [7].
Numerous other possibilities, in this direction of generalization exists with
respect to choice of geometry of these lattice points and it is to be expected
that the approximants resulting from considerations of various geometries
will have properties desirable in the synthesis of two-dimensional systems.
An important objective of the research was to investigate into the structure
and properties of these 2-D rational approximants, in addition to providing
fast methods for their computation.
The approximants are required to satisfy certain desirable properties,
when used to design complex systems. Two such properties are stability and
robustness to parameter variations. Another objective of the research was
to provide suitable means for incorporating these desirable features in the
approximants.
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7
The final objective in the approximation phase of the research was to
investigate into the conditions on the coefficients of the rational approximants
so that certain desirable properties like maximal flatness or equiripple behavior
in the frequency response of the designed multidimensional filters would be
obtfined.
ii. Multidimensional Signal Processing Over a Finite Field
The theory for I-D discrete systems which are linear over the finite
field of two elements (binary field), GF(2), is well established. One-
dimensional discrete systems which are linear with respect to the finite field
of two elements, 0 and 1, have been developed for applications such as
generation of pseudorandom sequence and error-correcting codes. Such systems
are characterized by rational functions and polynomials in one variable with
coefficients over GF(2). Recently, the need for study of properties of two
variable rational functions in the quotient field of the ring of bivariate
polynomials with coefficients belonging to GF(q), has been felt in the synthesis
of psuedornadom noise arrays that exhibit special correlation properties,
system identification, and two dinxnsional filtering over a finite field.
There is no satisfactory definition for primitivity of 2-D polynomials over
a finite field, though that concept has played a vital role in the 1-D case
especially in error-correcting code construction and generation of mximum
length shift register sequences, which in turn are ideally suited for the
identification of linear systems using cross-correlation techniques. For
bivariate polynomials, the study of primitivity, the generation and enumeration
of irreducible polynomials, the study of their corresponding array and their
autocorrelation properties was, therefore, considered to be potentially very
useful. An added advantage to this study, is the observation of the fact that
there is much in common between the algebraic structure that characterize
these and the approximation problem discussed earlier.
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8
iii. Primitive Factorization, Coprimeness, Matrix Fraction Description
Research in this area was considered to be very fundamental to the general
theme of research conducted.
The need for factoring a given polynomial matrix A(z) in the form A(z) =
AI(z) A2(z) (where the polynomial matrix factors A 1z) and A2(z) , besides
having elements in the same ring to which the elements of A(z) belong, might
also be required to satisfy certain special properties), occurs in many situations.
These include network theory, filtering of multiple time series, system theory,
damped vibrations, transport theory, and Wiener-Hopf equations. Special atten-
tion to spectral factorizations, where A1(z) and A2 (z) have their spectra located
one inside and the other outside a Cauchy contour with A(z) analytic on a neigh-
borhood of the contour, has been given in [8]. A more general type of factori-
zation theory occurs in the polynomial matrix approach to realization theory of
linear systems [9], which, incidentally, has close links to the state-space reali-
zation theory [10]. A state-space realization, S(z) = H(zt-F)- G of a matrix of
strictly proper rational functions has minimal dimension if and only if (F,G) is
controllable and (F,H) is observable. On the other hand, a polynomial realization,
S(z) = W(z) + V(z) [T(z)] U(z)
is irreducible if T(z), U(z) are relatively left prime and T(z), V(z) are rela-
tively right prime. The connection is provided by a theorem of Rosenbrock which
states that a pair of constant matrices (F,G) is controllable if and only if
the pair zI - F, G of polynomial matrices is relatively left prime.
Recent developments in multidimensional systems theory [11], [121, motivated
by a variety of practical applications [131 necessitated investigations of the
scopes for extension of the univariate (l-D) multivariable (or multi-input multi-
output) results of Rosenbrock and others to the n-D (n > 1) case. The realization
theory based on the 1-D polynomial matrix approach is valid for polynomial matri-
ces (generated, for example, from an initial non-irreducible realization of S(z))
with elements from an arbitrary principal ideal domain and when the solution
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9
(say, an irreducible realization) is to be constructed, the domain is further
required to be Euclidean. In n-D (n > 1) systEm theory, this is not the case.
In [14], Morf, Levy and Kung showed how the greatest common right divisor (gcrd)
could be extracted from two specified bivariate polynomial matrices based on
their primitive factorization theorem. The operations necessary to implement
the extraction scheme, however, required the ground field to be algebraically
closed. Subsequently, Youla and Gnavi [15] showed the infeasibility, in general,
of primitive factorization for polynomial matrices in three variables and among
other interesting results also presented a primitive factorization algorithm
for 2-D polynomial matrices similar to the one in [7]. This algorithm also
requires the ground field to be algebraically closed. This requirement is not
satiqfied in many problemsof practical interest. In continuous linear system
theory, the ground field is usually real, while in the discrete case it could
also be finite [13, ch. 6]. With the objective of increasing considerably the
scope for application of the important theoretical results in [14], [15], the
question as to whether or not the primitive factorization theorem, presented
in 114,], holds over any arbitrary, but fixed, field of coefficients was posed
in [13]. A very important objective of the research was to obtain an answer
to the question posed and explore the impact of the result to various applications
of interest to the Air Force.
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10
References
[11 R. E. Kalman, "On partial realization, transfer functions and canonical
forms," ACTA Polytechnica Scandinavica, (Mathematics and Computer Science)
Vol. 29, November 1979, pp. 3-32.
[2] J. Rissanen, "Algorithms for triangular decomposition of block Hankel and
Toeplitz matrices with application to factoring positive matrix polynomials,"
Math. Comp., Vol. 27, No. 121, January 1973, pp. 147-154.
(3] B. W. Dickinson, M. Morf, and T. Kailath, "Aminimal realization algorithm
for matrix sequences," IEEE Trans. Auto. Control, 10, Feb. 1974, pp. 31-38.
[4] C. Lanczos, "Solution of systems of linear equations by minimized
iterations," J. of Res. NBS, 49, July 1952, pp. 33-53.
[5] J. L. Massey, "Shift register synthesis and BCH decoding," IEEE Trans.
Information Theory, 15, Jan. 1969, pp. 122-127.
[6] J. S. R. Chisholm, "Rational approximants defined from double power
series," Math. of Computation, 27, Oct. 1973, pp. 841-848.
(7] B. Levy, M. Morf, and S. Y. Kung, "New results in 2-D system theory, part
3," presented at Midwest Symp. Circuits and Systems, Lubbock, Texas 1977.
[8] H. Bart, I. Gohberg and M. A. Kaashoek, "Minimal Factorization of Matrix
and Operator Function," Birkh;1user Boston Inc., Cambridge, MA, 1979.
[9] H. H. Rosenbrock, "State-Space and Multivariable Theory," Nelson, London,
1970.
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11
[10] R. E. Kalman, "Irreducible realizations and the degree of a rational
matrix," SIAM. J. Appl. Math., 13, 1965, pp. 520-544.
[il] N. K. Bose, Ed. "Multidimensional Systems," Proceedings of IEEE, 65, June
1977.
[12] N. K. Bose, Ed. "Multidimensional Systems: Theory and Applications," IEEE
Press, NY, 1979.
[13] N. K. Bose, "Applied Multidimensional Systems Theory," Van Nostrand
Reinhold Co., NY, 1981.
[14] M. Morf, B. C. Levy and S. Y. Kung, "New results in 2-D systems theory,
Part 1: 2-D polynomial matrices, factorization and coprimeness," Proc.
IEEE, 65, June 1977, pp. 861-872.
[15] D. C. Youla and G. Gnavi, "Notes on n-dimensional system theory," IEEE
Trans. Circuits and Systems, 26, Feb. 1979, pp. 105-111.
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c. Status of the Research
(i) Rational Approximants
Let K[z] be the set of formal power series over a ring K with indeterminate
-. z. In the present context, K will be the ring of constant rectangular
matrices of order m x n, whose elements do not necessarily commute.. The
matrix Pad6 approximation problem then consists of finding two matrix
polynomials AL(z) and BM (z) such that for a given T(z) E K[z]:
(1) 6[AL(z)] < L, 6IBM(z)] < M, where "6" denotes "degree"
(2) the residual,
R(z) = T(z) BM (z) - AL(z)
satis fies
ord R(z) > M + L + 1
In the conventional scalar Pad6 theory, for AL(z) and BM(z) to be devoid
of a common divisor of degree greater than zero, it is necessary that
B M(0) be nonzero. In the case considered here, B M(0) is taken to be
nonsingula and withot loss of generality it may be set equal to the
identity matrix I of appropriate order.
B M(0) = I
AL(z) BM-1(z) will then denote the right matrix Pad6 approximant
of order [L/M]. The LMPA (L stands for left) may be (RMPA) analogously
defined. In [c.l], it has been established that (except possibly in
certain derogatory cases), a recurrence relation relates the [L+I/M+I],
[L/M], and [L-l/M-l] order matrix Pade approximantq (left or right), the
existence of which is guaranteed from the vailidity of a certain rank
condition on the corresponding set of characterizing matrices possessing
a block-Hankel structure.
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13
The preceding results have been used to define and study a special type
of power series belonging to the ring of constant square matrices. The main
result documented in [c.2] is the proof of the relationship between rational
aoproximants of appropriate orders to a specified symmetric matrix power series
of the special type referred to and mulitport network synthesis using the matrix
version of the classical continued fraction expansion theory and the recently
developed artifice of matrix Cauchy index [c.3]. The "denominator" polynomial
matrices of the approximants are shown to form an orthogonal polynomial matrix
sequence over a real semi-infinite interval. Though mathematical derivations
of the properties of the polynomial matrices belonging to the orthogonal poly-
nomial matrix sequence have been given, the reader has also been alerted to
the feasibility of network-theoretic justification of the results. It is
hoped that these links between mathematical results and the theory of network
realizability will kindle interest for further research among scientists coming
from either disciplines.
The mathematical literature [c.41, [c.51, contains some results on the
extension of the Padd approximation scheme to power series in more than one
indeterminate, primarily with a veiw towards applications in problems of physics
and numerical analysis. In these approaches, however, the Hankel nature of
the characterizing matrix is lost and, therefore, the feasibility of recursive
computation is seriously impaired. In the contexts of two-dimensional digital
filtering, stochastic realization and image processing, the speed of computa-
tion is crucial. In order to conform to this requirement, the questions of
existence, nonuniqueness alongwith scopes for recursive computation when
existence is guaranteed, are investigated in depth in (c.61. In [c.61, attention
is directed to the exploitation of a block Hankel-Hankel matrix structure
(where the elenments of the block may also be viewed as block-Hankel matrices)
that can be used to characterize the problem. A sequence of vertical and
horizontal Pad6 approximants over the chosen "Hankel matching grid" are defined.
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14
A three-term recurrence formula is developed to compute, first, all the vertical
approximants of a prescribed order, in the sequence. In physical problems, the
rational approximants are required to be stable. A sufficient condition for
BIBO stability based on the zero distribution of a bivariate polynomial,
obtained directly via computation of the determinant of a matrix formed from
the input data, augmented by a row of matrix indeterminates, has been obtained
in (c.7]. Thus, if one is willing to invest one's time in the computation of
the determinant of a specified matrix followed by a check for stability of one
bivariate polynomial (the determinant computed), the effort required to actually
calculate an approximant, which will turn out to be unstable, may be saved.
For generalization of the concept of Pad6 approximants to nonlinear operators
and its scopes for tackling nonlinear systems of equations, nonlinear initial
and boundary value problems, nonlinear partial differential equations and
nonlinear integral equations, a recent dissertation [c. 8 ] is of interest.
Since 1-D techniques for computing Pad6 approximants are more well developed
and naturally, faster, the feasibility of constructing 2-D rational approximants
to match exactly a prescribed set of coefficients in a double power series, by
using only the 1-D Pad4 technique has been explored in [c.91. The matching grid
is not restricted to a rectangular grid though for brevity in exposition this
type of grid is focussed upon in [c.9]. There, it is shown that it is possible
to obtain a 2-D rational approximant by computing several 1-D Pad6 approximants
over power series coefficients that belong to a field and only one I-D Pade
approximant over power series coefficients that belong to the field of rational
functions in I indeterminate. The procedure and basic philosophy extends easily
to the n-D case, n > 2. The speed and flexibility in implementation of the
scheme is likely to be useful in various types of filter design problems.
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15
Positivity and Stability
Positivity and stability results, which are fundamental in the area of
multidimensional systems design were established on firm foundations. It has
already been pointed out that rational approximants are useless for implementa-
tion, if unstable. A new stability test for n-D digital filters has been obtained
in [c.lO]. This test dwells on a conceptually different philosophy than other
algebraic tests advanced for multidimensional filters, as noted in [c.ll], though
the computational advantages in implementation may not be present, in general.
The problem of testing a polynomial for zeros on a polydisc distinguished
boundary is considered in[c.121. Such tests are required in testing a linear
discrete shift-invariant processor for exponential stability [c.131 and in
theuse of ceptral techniques [c.1 41, the discrete Hilbert transform and
homomorphic filtering. The test procedures for stability often dwell on the
checking of a multivariate polynomial for local or global positivity. It is
shown in [c.151 that certain singular cases occurring in the test of a multi-
variate polynomial for global positivity need not be considered, leading to
a considerable improvement in the efficiency in implementation of the test
procedure.
Robustness
In problems originating from physical systems, the multivariate polynomials
characterizing some properties of the system under study, like stability or
existence of limit cycles, might not have integer coefficients and might even
possess coefficients which could take arbitrary values over specified intervals.
In such cases, it is useful to know the allowable interval within which the
coefficients might fluctuate centered around their respective unperturbed
Values, so that the test implemented on the polynomial with these unperturbed
coefficient values, if found to yield the positivity property will guarantee
invarLance of the positivity property under perturbation of any or all of the
hebb..
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16
coefficients within the allowable interval. In [c.16], it is first shown
how the largest symmetric interval centered around each coefficient of a globally
positive univariate polynomial may be determined so that the positivity
property is invariant under any perturbation of the coefficients within that
interval (or its integer multiple). A characterization theorem for the deter-
mination of a similar interval is given for multivariate polynomials and a
scheme for finding the interval via recursive computation of resultants, allow-
ing the elimination of at least one indeterminate at each step, is given. This
result partially contributers towards the resolution of the multidimensional
filter stability problem under coefficient fluctuation in the characterizing
denominator polynomial of the filter due to any sources of error -- roundoff,
quantization etc. In [c.17], the main goal of obtaining conditions for
invariance of stability properties as directly as possible in term of the
original coefficients (just as was done with respect to the global positivity
property in [c.161) was obtained for low degree polynomials. Though, higher
degree polynomials are not tackled in [c.17], the results should be useful
from the practical standpoint, since often the design of higher order systems
is based on the design of low order blocks or prototypes.
Stabilization
Following the derivation of a rational approximant and implementation of
the stability test a filter may be found to be unstable. A practical problem
encountered, then, is to stabilize the filter without appreciable change in
the magnitude of the frequency response. Two approaches towards the attain-
ment of this goal are the planar least squares Inverse (PLSI) approach and the
discrete Hilbert transform (DIIT) approach. In [c.181 it is shown using recent
results from algebraic computational complexity theory that the multiplicative
complexity of computation of a 2-D DHT is not greater than twice the sum of
multiplicative complexities of two 1-D DHT's.
-
(ii) Multidimensional Signal Processing Over a Finite Field
An important fact exploited in many applications (like the construction
of error correcting codes) is the existence of irreducible single variable
polynomials of any specified degree whose coefficients belong to the finite
field, GF(q) (the binary field case, where q=2, is of greatest practical
significance). In [c.19], a nontrivial extension of a known procedure to
obtain the number of distinct irreducible monic univarate polynomials of
prescribed degree m with coefficients in GF(q) is made with respect to
multivariate polynomials. Specifically, a procedure is given to determine the
number of distinct irreducible polynomials of prescribed degree in each of the
independent variables and having coefficients in GF(q).
In [c.20], an analysis of the structure and properties of the impulse response
array of 2-D discrete space systems, characterized by rational functions
having coefficients in GF(q), is given. It is shown that such an array exhibits
a rwo (column) type of periodicity. This property specializes to the doubly
periodic property if and only if the denominator polynomial of the transfer
function is separable into a product of univariate polynomials. Arrays with
a maximum of three levels in the autocorrelation function are identified and
explicit expressions for these levels are given. This result provides the
complete characterization of arrays which are remarkably close to the pseudo-
random arrays sought in a variety of applications including the study of
spectrometric imagers and sound diffusion.
(iii) Primitive Factorization, Coprimeness, Matrix Fraction Description
A problem of importance both from the theoretical and application viewpoints
has been solved during the course of this research. In 1977 and 1979, algorithms
were presented to extract in some sense the content of a full rank matrix A with
entries in the ring K[z,w] of bivariate polynomials over some field K. However,
k.J
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"18
the algorithm presented in both cases specify and require the coefficient
field K to be algebraically closed - typically the field of complex numbers.
It is desirable, from theoretical and computational standpoints to have no
such restriction on K; so, for example, one could do the factorization over
the real field or even the field of rational numbers, provided the coefficients
start out in these field. In [c.21] an algorithm which produces a primitive
factorization over an arbitrary field K is presented. Several related results
leading to a general factorization theorem are also stated and proved.
References
(c.l] N. K. Bose and S. Basu, "Theory and recursive computation of 1-D
matrix Pad6 approximants," IEEE Trans. on Circuits and Systems,
27, April 1980, pp. 323-325.
[c.2] S. Basu and N. K. Bose, "Matrix Stieltjes series and network models,"
SIAM J. Math. Analysis, 14, March 1983, pp. 209-222.
[c.3] R. Bitmead and B. D. 0. Anderson, "The matrix Cauchy index: properties
and applications," SIAM J. Appl. Maths, 33, Dec. 1977, pp. 655-672.
[c.4] J. S. R. Chisholm and J. Mewan, "Rational approximants defined from
power series in n variables," Proc. Royal Soc., A336, 1974, pp. 421-
452.
(c.51 D. Levin, "General order Padd type rational approximants defined from
a double power series," J. Inst. Maths. Applics., 18, 1976, pp. 1-8.
(c.6] N. K. Bose and S. Basu, "Two-dimensional matrix Pad6 approximants:
existence, nonuniqueness, and recursive computation," IEEE Trans.
Auto. Control, 25, June 1980, pp. 509-514.
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19
[c.7] S. Basu and N. K. Bose, "Stability of 2-D matrix rational approximants
from input data," IEEE Trans. Auto. Control, 26, April 1981, pp. 540-541.
[c.8] Annie A. M. Cuyt, "Abstract Padg Approximants and Operators:: Theory and
-. Applications," Doctoral Dissertation, Department of Mathematics,
University of Antwerp, Belguim, 1982.
[c.9] N. K. Bose, "2-D rational approximants via I-D Pad4 technique,"
Signal Processing: Theories and Applications ed. by M. Kunt and
F. de Coulon, North-Holland Publ. Co., EURASIP, 1980, pp. 409-412.
[c.10] N. K. Bose, "Implementation of a new stability test for n-D filters,"
IEEE Trans. Acoustics, Speech, and Signal Proc., 27, Feb. 1979, pp. 1-4.
[c.ll] T. S. Huang, ed., "Two-Dimensional Digital Signal Processing 1",
Springer-Verlag, Topics in Applied Physics Series, vol. 42,
New York 1981.
[c.121 N. K. Bose and Sankar Basu, "Tests for polynomial zeros on a polydisc
distinguished boundary," IEEE Trans. Circuits and Systems, 25,
Sept. 1978, pp. 684-693.
[c.13] S. L. Chang, "Digital linear processor theory and optimum multidimensional
data estimation," IEEE Trans. Auto. Control, 24, April 1979, pp. 199-201.
[c.14] R. M. Mersereau and D. E. Dudgeon, "Two-dimensional digital filtering,"
Proc. IEEE, 63, April 1975, pp. 610-623.
[c.151 N. K. Bose, "Multivariate polynomial positivity test efficiency
improviement," Proc. IEEE, 67, October 1979, pp. 1443-1444.
[c.161 N. K. Bose and J. P. Guiver, "Multivariate polynomial positivity
invariance under coefficient perturbation," IEEE Trans. Acoustics,
Speech, and Signal Proc., 28, Dec. 1980, pp. 660-665.
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[c.17] J. P. Guiver and N. K. Bose, "Strictly Hurwitz property invariance of
quartics under coefficient perturbation," IEEE Trans. Auto. Control,
28, January 1983, pp. 106-107.
[c.18] N. K. Bose and K. A. Prabhu, "Two-dimensional discrete Hilbert transform
and computational complexity aspects in its implementation," IEEE
Trans. Acoustics, Speech, and Signal Proc., 27, August 1979, pp.
356-360.
(c.19] K. A. Prabhu and N. K. Bose, "Number of irreducible q-ary polynomials
in several variables with prescribed degrees," IEEE Trans. Circuits
and Systems, 26, Novermber 1979, pp. 973-975.
[c.20) K. A. Prabhu and N. K. Bose, "Impulse response arrays of discrete-space
systems over a finite field," IEEE Trans. Acoustics, Speech, and Signal
Proc., 30, February 1982, pp. 10-18.
[c.211 J. P. Guiver and N. K. Bose, "Polynomial matrix primitive factorization
over arbitrary coefficient field and related results," IEEE Trans.
Circuits and Systems, 29, October 1982, pp. 649-657.
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d. Publications in Technical Journals
1. N. K. Bose and S. Basu, "Tests for polynomial zeros on a polydise distinguished
boundary," IEEE Trans. on Circuits and Systems, vol. 25, Sept. 1978
-° (Special Issue on Mathematical Foundations of Systems Theory), pp. 684-693.
2. N. K. Bose, "Implementation of a new stability test for n-D filters,"
IEEE Trans. Acoustics, Speech and Signal Processing, vol. 27, February, 1979,
pp. 1-4.
3. N. K. Bose, "A result on the use of modular methods in multidimensional
computations," Archiv fUr Elektronik und Ubertragungstechmik, Band 33, 1979,
pp. 213-218.
4. N. K. Bose and K. A. Prabhu, "Two-dimensional discrete Hilbert transform
and computational complexity aspects in its implementation," IEEE Trans.
Acoustics, Speech, and Signal Processing, vol. 27, Aug. 1979, pp. 356-360.
5. N. K. Bose, "Multivariate polynomial positivity test efficiency improvement,"
Proc. of the IEEE, vol. 67, Oct. 1979, pp. 1443-1444.
6. K. A. Prabhu and N. K. Bose, "Number of irreducible q-ary polynomials
in several variables with prescribed degrees," IEEE Trans. Circuits and
Systems, vol. 26, November 1979, pp. 973-975.
7. N. K. Bose and S. Basu, "Theory and recursive computation of 1-D matrix
Padg approximants," IEEE Trans. Circuits and Systems, vol. 27, April 1980,
pp. 323-325.
8. N. K. Bose and S. Basu, "Two-dimensional matrix Padd approximants: existence,
nonuniqueness, and recursive computation," IEEE Trans. Auto. Control, vol.
25, June 1980, pp. 509-514.
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9. N. K. Bose and J. P. Guiver, "Multivariate polynomial positivity invariance
under coefficient pertubration," IEEE Trans. Acoustics, Speech, and
Signal Proc., vol. 28, Dec. 1980, pp. 660-665.
10. S. Basu and N. K. Bose, "Stability of 2-D matrix rational approximants
from input data," IEEE Trans. Auto. Control, vol. 26, April 1981, pp.
540-541.
11. J. P. Guiver and N. K. Bose, "On test for zero-sets of multivariate
polynomials in noncompact polydomains," Proc. IEEE, vol. 69, April 1981,
pp. 467-469.
12. K. A. Prabhu and N. K. Bose, "Impulse response arrays of discrete-space
systems over a finite field," IEEE Trans. Acoustics, Speech, and Signal
Proc., vol. 30, Feb. 1982, pp. 10-18.
13. J. P. Guiver and N. K. Bose, "Polynomial matrix primitive factorization
over arbitrary coefficient field and related results," IEEE Trans.
Circuits and Systems, vol. 29, October 1982, pp. 649-657.
14. J. P. Guiver and N. K. Bose, "Strictly Hurwitz property invariance of
quartics under coefficient perturbation," IEEE Trans. Auto. Control,
vol. 20, January 1983, pp. 106-107.
15. S. Basu and N. K. Bose, "Matrix Stieltjes series and network models,"
SIAM J. Math. Analysis, vol. 14, March 1983, pp. 209-222.
16. H. M. Valenzuela and N. K. Bose, "Maximally flat rational approximants
in multidimensional filter design," Circuits, Systems, and Signal
Processing, vol. 2, #1, 1983.
4.9
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17. J. P. Guiver and N. K. Bose, "Causal and weekly causal 2-D filters with
applications in stabilization," chapter 3 of "Recent Advances in Multidi-
mensional Systems" by N. K. Bose (to appear).
BOOKS
1. N. K. Bose, editor "Multidimensional Systems: Theory and Applications,"
IEEE Press, New York, 1979.
2. N. K. Bose, "Applied Multidimensional Systems Theory," Van Nostrand
Reinhold Co., N.Y., 1982.
4.
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24
e. List of the Professional Personnel
Associated with the Research Effort
1. K. A. Prabhu
_Completed his Ph.D. dissertation entitled, "Two-dimensional digital
filtering in a finite field." Was awarded the Ph.D. degree in Electrical
Engineering from the University of Pittsburgh in 1980. Currently employed
by Bell Laboratories, Holmdel, New Jersey, 07733.
2. S. Basu
Completed his M.S. thesis entitled "Theory and application of a direct
test procedure for polynomial zeros on polydisc distinguished boundary."
Was awarded the M.S. degree in Electrical Engineering from the University
of Pittsburgh in 1978.
Subsequently, completed his Ph.D. thesis entitled, "Multivariable one-
and two-dimensional Padd approximation theory and its appli-itions."
Was awarded the Ph.D. degree in Electrical Engineering from the
University of Pittsburgh in 1980. Currently, serving as an Assistant
Professor in the Department of Electrical Engineering, Stevens Institute
of Technology, Hoboken, New Jersey, 07030.
3. John P. Guiver
Completed his Ph.D. dissertation on "Contributions to two-dimensional
systems theory." Was awarded the Ph.D. degree in Mathematics from the
University of Pittsburgh in 1982. Currently serving as a Guidance and
Control Engineer with British Aerospace, at Bristol, England.
4. Hector M. Valenzuela
Completed his M. S. in December, 1980. Worked on research in the area of
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25
"Maximally flat rational approximants in multidimensional filter design,"
and used this research to meet the requirements of Ph.D. Preliminary
Examination in Electrical Engineering in September, 1981. Was admitted
as a candidate for the Ph.D. degree in Electrical Engineering in
November, 1982. Is expected to complete all requirements for the Ph.D.
degree by October, 1983. His Ph.D. dissertation will be in the area
of "Shift-Variant multidimensional systems," which research is a continuation
of research efforts in multidimensional linear shift-invariant systems.
5. John Loney
Worked in the very initial stages of research in rational approximation
theory, for a period of one month only.
6. Robert Wen
Implemented some examples required by the Principal Investigator, on the
computer.
7. H. M. Kim
Beginning his Ph.D. studies in Electrical Engineering following the
completion of his M.S. in 1982. Is being trained to serve as a graduate
student researcher to the Principal Investigator following the completion
of Hector M. Valenzuela's Ph.D. dissertation.
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26
f. Interactions
I. Seminars or talks
1. N. K. Bose, "Test for polynomial zeros and consequences," presented at the
International Symp. on Circuits and Systems at New York, May 1978.
2. N. K. Bose, "2-D matrix Pad4 approximants - existence, nonuniquenss and
recursive computation," invited talk at the 4th International Symp. on
Mathematical Theory of Networks and Systems, Delft, Holland, on July 3,
1979.
3. N. K. Bose, "Multivariate positivity," invited talk at I.I.T, New Delhi,
India, on July 6, 1979.
4. N. K. Bose, "Aspects of 2-D systems theory: stability and positivity,"
invited talk at Tohoku University, Sendai, Japan, on July 16, 1979.
5. N. K. Bose, "2-D discrete Hilbert transform and computational complexity
aspects in its implementation," invited talk in the Special Session on
Computational Complexity at the 1979 International Symposium on Circuits
and Systems, Tokyo, Tapan, on July 17, 1979.
6. N. K. Bose, "Multidimensional Padd approximation theory," invited talk
in the Special Session on Distributed and Multivariable Networks in the
1979 Interantional Symposium on Circuits and Systems, Tokyo, Japan on
July 18, 1979.
7. N. K. Bose, "Algebraic methods for stability testing," invited talk at
the First Acoustics, Speech, and SIganl Processing Workshop on Two-
Dimensional Digital Signal Processing at Lawrence Hall of Sciences,
Berkeley, California on October 3, 1979.
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27
8. N. K. Bose, "Mathematical realizability theory," invited talk at the
Carnegi Mellon University and University of Pittsburgh Joint Colloquium,
Department of Mathematics, Pittsburgh, on February 8, 1980.
9. N. K. Bose, "Aspects of two-dimensional digital signal processing,"
invited talk at the Carnegie-Mellon University Colloquium in Electrical
Engineering, Pittsburgh, on March 20, 1980.
10. N. K. Bose, "Finite field methods in multidimensional systems theory,"
invited talk in a Special Session on Recent Advances in Mathematical
System Theory at the International Symposium on Circuits and Systems,
Houston, Texas, April 29, 1980.
11. S. Basu and N. K. Bose, "Matrix Stieltjes series and network models,"
presented by S. Basu at the Midwest Symposium on Circuits and Systems, Toledo,
Ohio, on August 5, 1980. Also presented by N. K. Bose as an invited talk
at Department of Mathematics, University of Antwerp on June 17, 1982
and at Gesamthochschule Wupper .ii )n Junt 24, 1982.
12. K. A. Prabhu and N. K. Bose, "Impulse response arrays of discrete-space
systems over a finite field," presented by N. K. Bose at the 1981 Inter-
national Symposium on Circuits and Systems, Chicago, Illinois, April
28, 1981.
13. N. K. Bose, "Topics in multidimensional systems theory," invited talk at
University of Illinois, Chicago, Illinois, on January 21, 1982.
14. N. K. Bose, "Polynomial matrix primitive factorization over arbitrary
coefficient field and related results," at Phillips Research Lab.,
Brussels on June 18, 1982, at Department of Mathematics, Vrije
Universiteit, Amsterdam on June 21, 1982, at Department of Mathematics,
Technische Hogeschool, Eindhoven on June 23, 1982 and at Institute of
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28
of Automatic Control and Industrial Electronics at ETH, Zurich on
June 20, 1982.
15. N. K. Bose "Bidimensional digital filtering over a finite field -
latest results," invited seminars at Laboratory on Digital Signal Processing,
Swiss Federal Institute, Lausanne, on June 25, 1982 and at Institute of
Automatic Control and Industrial Electronics, ETH, Zurich on June 29, 1982.
16. N. K. Bose "Applications of 2-D systems theory: spatio-temporal filtering
over finite and infinite fields," invited seminar at Department of
Mathematics, Technische Hogeschool, Eindhoven June 22, 1982.
17. N. K. Bose was invited to give four seminars on recent results in multi-
dimensional systems theory including "stabilization of two-dimensional
feedback systems by causal and weakly causal compensators," at Department
of Electrical Engineering, University of Waterloo on July 12, July 13,
July 15 and July 16, 1982.
II. Professional Recognitions and Activities
1. Regular reviewer for Zentralblatt fUr Mathematik and Mathematical Reviews.
2. Regular reviewer of papers submitted to journals including: Proceedings of
IEEE, IEEE Transactions on ASSP, CAS, and Automatic Control, Automatica,
Int. J. on Control, J. of the Franklin Institute, and Canadian Electrical
Engineering Journal.
3. Elected to be Fellow of IEEE for contributions to multidimensional
systems ther-y and circuits and systems education, effective January 1, 1981.
4. Served as Associate Editor, IEEE Transactions on Circuits and Systems
between June 1980 to June 1981.
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5. Serving as Chairman of Education Committee, IEEE Society on Circuits
and Systems, since 1979.
6. Serving as member of steering committee on Midwest Symp. on Circuits and
Systems, since 1969.
7. Invited to chair a Session on "Large-scale dynamical systems," at the
First International Conference on Circuits and Computers, Port Chester,
New York, on October 2, 1980.
8. Organized a Pre-Symposium Workshop on "Applied Multidimensional Systems
Theory," on April 26, 1981 preceding the 1981 International Symposium on
Circuits and Systems, held at Chicago, Illinois between April 27, 1981 to
April 29, 1981. The workshop was very well attended by participants, both
from industries and universities, and was considered to be highly successful.
9. Invited to participate in a workshop on Rational Approximations for Systems
at Leuven, Belgium on August 31 - September 1, 1981.
10. Invited to be a member of the Program Committee of the European Conference
on Circuit Theory and Design at the 1981 European Conference on Circuit
Theory and Design, held at the Hague, Netherlands, August 25-28, 1981.
11. Invited to be a member of the Technical Program Committee, in charge of
systems at the International Symposium on Circuits and Systems, to be
held at Newport Beach, California, May 2-4, 1983.
12. Reviewer of research proposals submitted to the Engineering Division of
National Science Foundation on August 1978, June 1980, and July 1982.
13. Reviewer of proposal submitted to the Division of International programs
of National Science Foundation on January 1980 (U.S./Australian Joint
Seminar/Workshop) and January 1983 (US-West European Cooperative Science
Programs).
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30
14. Reviewer of research proposal submitted to the Air Force.
15. Invited to serve on the Technical Program Committee in the area of Multi-
dimensional Processing at the forthcoming International Conference on
-.Acoustics, Speech and Signal Processing to be held in San Diego, California,
March 19 - 21, 1984.
16. Invited to write a monograph on Recent Advances in Multidimensional System
Theory by the Managing-Editor, Professor Dr. M. flazewinkel of Mathematics
and Its Applications, D. Reidel Publishing Company.
17. Invited to edit a Special Issue on "Spatio-Temporal Filtering," to be
published by Circuits, Systems, and Signal Processing, Birkh~user Boston Inc.
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31
g. Specific Applications Stemming from Research Report
1. The Pad6 type of rational approximants occur in the problem of partial
realization of systems from prescribed input-output map. Tlese approximants
are.also known to be closely linked to Prony's method, whose system theoretic
applications include its use in the identification of linear dynamic systems
from time-domain measurements. Apart from the system theoretic relevance
of Pad6 theory, other engineering applications include time domain design of
digital filters and order reduction of control systems. In the problem of
digital filter design the coefficients of the power series correspond to the
impulse response sequence of the filter. The problem, then, is to find via
the Pad4 method a realizable transfer function or matrix (in the multivariable
or multivariable bidimensional cases, which have been the prime targets in
the initial stages of this research) from a finite section of the impulse
response characteristics. In the context of order reduction of control systems,
the objective is to find a transfer function or matrix of lower order than
the original transfer function or matrix such that the suitable predetermined
number of Markov parameters of the lower order system match with those of
the original one.
2. The investigation into the structure and properties or matrix polynomials,
orthogonal over a real interval, has been considerably illuminated by the
physical insights offered from the well developed theory of passive linear
multiport network synthesis, to which the mathematical topic is shown to be
closely linked. The developed topic of matrix orthogonal polynomials
over a real line is the first comprehensive treatment of the question and its
subsequent influence in multichannel continuous system theory (the multi-
channel discrete system theory and matrix orthogonal polynomials over an
unit circle were topics investigated in 1978 by researchers in U.S.A. and
Europe).
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32
3. Since, in physical problems, an approximant is required to be stable,
algebraic tests for stability were fully developed. As a result, these
algebraic tests for verifying whether or not a multidimensional linear shift-
invariant filter is stable are now well established and, if desired, can be
imptemented with infinite precision, provided adequate computational resources
are available. Recognizing the computational chore needed to test for
stability, in the 2-D setting, a criterion for stability of a 2-D matrix rational
approximant over a Hankel grid was developed, based on the input data. The
criterion yields a sufficient condition for BIBO stability.
4. Contributions to the problem of stability invariance under coefficient
fluctuation within a symmetric interval centered around their original values
are of direct interest when attempting to resolve the multidimensional digital
filter stability problem under coefficient fluctuation due to any sources of
error-roundoff, quantization etc, as well as in the study of linear active
analog filter stability properties when the objective is to determine the allow-
able ranges of variation of one or more parameters (like gains of op-amps
imbedded in the filter), without affecting system stability. Therefore, the
foregoing contributions will be useful in the incorporation of nonlinearity in
models used in multidimensional filter stability investigations.
5. The results obtained on the zero-sets of multivariate polynomials in non-
compact polydomains will find use in multidimensional discrete-continuous
stability investigations including, for example, the study of asymptotic
stability, independent of delay, in differential equations of the neutral type.
6. The study of the periodicity, autocorrelation and crosscorrelation properties
of the matrix impulse response sequence associated with a transfer matrix having
coefficients in Zq, and obtained results on means to construct maximal length
matrix impulse response sequences via application of feedback on the state model
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33
constructed from the specified transfer matrix or otherwise, are expected to
be useful for simultaneous ranging to several targets, multiple terminal
system identification, and possibly code-division multiple access communication
systems.
7. A consequence of the primitive factorization algorithm developed is that
the gcd extraction algorithm, general factorization theorem, equivalence of
minor and factor left (or right) coprimeness concepts in the 2-D case go
through with no restriction on the coefficient field K.
8. The primitive factorization theorem enables us to construct the desired
factors not only when the elements of a given matrix are in D[w] with D = K[z]
but also when D is any Euclidean domain. Let K c(z) and K (z) denote,
respectively, the rings of proper real rational functions and stable proper
real rational functions. Then, a rational matrix T(z,w) of order m x 2(m < Z)
with entries in K c(z)(w) or K (z)(w) has a matrix fraction description, which
after the application of the primitive factorization theorem can be written in
the form,
T(z,w) = T7I (z,w) T2 (z,w)
where the matrices TI(z,w), T2 (z,w) each have entries in Kc (z)[w] or Ko (z)(w]
and are relatively left coprime there. It may be possible to set up a synthesis
scheme based on these coprime factors, which might yield a realization of lower
overall dimension in the dynamic elements z and w than is feasible via Eising's
method.' It is of interest to note that the lossless positive real property
is characterizable by the factors in the matrix fraction description of a square
transfer matrix with entries in K(z), where K, here, is the field of real numbers.
With the availability of the primitive factorization theorem here, it may be
possible to extend these ideas to characterize as well as synthesize 2-D
matrices representing various types of systems including 2-D discrete space
systems (causal and weakly causal as well as differential dealy systems and
lumped-distributed networks).
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34
9. The following result has been proved: A multivariable 2-D plant P(z,w)
with a left coprime matrix fraction description, P(z,w) = DL I(z,w) N (z w) is
thstabilizable by means of a causal compensator if and only if the m order
miriqrs of [DL NL I have no common zeros in the closed unit bidisc, U- 2 .
The 2-D setting provides a nice theoretical framework for generalization of
these stabilization results to weakly causal systems. Among other results, it
has been shown that many causal system= which are not stabilizable by
means of a causal compensator can be stabilized by means of a weakly causal
compensator. (Here stabilizable is meant in the control sense - i.e. thenz ,w)
feedback system is stable). In fact given a scalar plant d(zw) with d(O,O)
0, n(0,0) = 0 one can achieve a structurally stable feedback system by
means of a causal compensator if and only if n(z,w) and d(z,w) have no common
zeroes in the unit bidisc, whereas if one allows a weakly causal compensator
it is only necessary that n(z,w) and d(z,w) have no common zeroes on the
distinguished boundary of the unit bidisc. Similar results hold in the multi-
input/multi-output or multi-variable case.
10. Explicit conditions required to be satisfied by the numerator and denominator
polynomial coefficients of a magnitude-squared rational response function,
IS 2 1 (jW1 , jw 2 )12 , so that it might approximate an ideal low-pass characteristic
along any straight line in the plane through (0,0) in a maximally flat manner,
have been obtained. The conditions for attenuation along all radial directions
through (0,0) is expressed in terms of a positivity of a form in two variables.
The technique is easily extended to the multidimensional case and can be used
to design filters other than those of the low-pass type. The theory could
be used to design lumped-distributed and variable-parameter filters.
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DATE,
ILMED'